Wikipedia
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions.
Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses on which Geometry is based). It is a very broad and abstract generalization of the differential geometry of surfaces in R. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled Einstein's general relativity theory, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.
Usage examples of "riemannian geometry".
But on the day after he had refused it for the first time, the memory banks decanted a double dose of projective Riemannian geometry, and he awoke to find four monitors holding him down on the couch during the last throes of a classical Jacksonian seizure.
Newton's laws, yes but the Hamiltonian, Riemannian geometry, wave functions?